Bases of reproducing kernels in model spaces.

Fricain, Emmanuel

Document type :

Compte-rendu et recension critique d'ouvrage

Title :

Bases of reproducing kernels in model spaces.

Author(s) :

Fricain, Emmanuel [Auteur]

Laboratoire Paul Painlevé - UMR 8524 [LPP]
Université de Lille

Journal title :

Journal of Operator Theory

Pages :

517–543.

Publisher :

Theta Foundation

Publication date :

2001

ISSN :

0379-4024

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

This paper deals with geometric properties of sequences of reproducing kernels related to invariant subspaces of the backward shift operator in the Hardy space $H^2$. Let $\Lambda=(\lambda_n)_n$ be a sequence in the unit ...
Show more >This paper deals with geometric properties of sequences of reproducing kernels related to invariant subspaces of the backward shift operator in the Hardy space $H^2$. Let $\Lambda=(\lambda_n)_n$ be a sequence in the unit disc $\mathbb D$, $\Theta$ be an inner function in $H^\infty(L(E))$, where $E$ is a finite dimensional Hilbert space, and $(e_n)_n$ a sequence of vectors in $E$. Then we give a criterion for the vector valued reproducing kernels $(k_Θ (., λ_n)e_n)_n$ to be a Riesz basis for $K_Θ:=H^2(E)\ominus \Theta H^2(E)$. Using this criterion, we extend to the vector valued case some basic facts that are well-known for the scalar valued reproducing kernels. Moreover, we study the stability problem, that is, given a Riesz basis $(k_Θ (., λ_n))_n$ , we characterize its perturbations $(k_Θ (., µ n))_n$ that preserve the Riesz basis property. For the case of asymptotically orthonormal sequences, we give an effective upper bound for uniform perturbations preserving stability and compare our result with Kadeč's 1/4-theorem.Show less >

Language :

Anglais

Popular science :

Non

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

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